Humphrey Stafford, 6th Earl of Stafford, 1st Duke of Buckingham, was the son of Edmund Stafford, 5th Earl of Stafford and Anne of Gloucester.
Anne of Gloucester was the daughter of Prince Thomas of Woodstock, the son of King Edward III.
Would Humphrey Stafford have insisted on his royal claim?
He might, but he has a weaker claim than the house of York.
All claims can be traced back to the sons of Edward III.
House of Lancaster: John of Gaunt, duke of Lancaster, who was the third son of Edward III.
House of York: Lionel of Antwerp, duke of Clarence, who was the second son of Edward III (maternal line) & Edmund of Langley, duke of York, who was the fourth son of Edward III (paternal line). (The great-granddaughter of Lionel of Antwerp married a son of Edmund of Langley.)
House of Stafford: Thomas of Woodstock, duke of Gloucester, who was the fifth son of Edward III (his daughter married into the house of Stafford).